The symmetric statistic k_(r,s,...) defined such that


where kappa_r is a cumulant. These statistics generalize k-statistic and were originally called "generalized k-statistics" (Dressel 1940). The term "polykay" was introduced by Tukey (1956; Rose and Smith 2002, p. 255). Polykays are commonly defined in terms of power sums, for example


Polykays can be computed using PolyK[{r, s, ...}] in the Mathematica application package mathStatica.

See also

k-Statistic, Polyache

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Cook, M. B. "Bi-Variate k-Statistics and Cumulants of Their Joint Sampling Distribution." Biometrika 38, 179-195, 1951.Dressel, P. L. "Statistical Seminvariants and Their Setimates [sic] with Particular Emphasis on Their Relation to Algebraic Invariants." Ann. Math. Stat. 11, 33-57, 1940.Rose, C. and Smith, M. D. Mathematical Statistics with Mathematica. New York: Springer-Verlag, 2002.Tukey, J. W. "Keeping Moment-Like Computations Simple." Ann. Math. Stat. 27, 37-54, 1956.

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Cite this as:

Weisstein, Eric W. "Polykay." From MathWorld--A Wolfram Web Resource.

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